Group homomorphism always exist between two groups.

Question

Is there always exists a homomorphism between two groups $G_1$ and $G_2$? Why?

Answer 1

If you require the homomorphism to be surjective, the answer is no; for example, there is no homomorphism of $\Bbb Z/3\Bbb Z$, the cyclic group of order $3$, onto $\Bbb Z/2\Bbb Z$, the cyclic group of order $2$.

Added: In particular, if $H:G_1\to G_2$ is a surjective homomorphism, then $$|G_2|=\frac{|G_1|}{|\ker h|}\;,$$ so $|G_2|$ must divide $|G_1|$. Another general class of examples for which the answer is no is provided by pairs of non-isomorphic groups of the same cardinality, like $\Bbb Z/4\Bbb Z$, the cyclic group of order $4$, and $(\Bbb Z/2\Bbb Z)^2$, the Klein four-group: a surjective homomorphism between them would be an isomorphism.

If you do not require the homomorphism to be surjective, the answer is yes: just map every element of $G_1$ to the identity element of $G_2$.